Abstract:We study the win rate $R_{N_d}/N_d$ of a biased simple random walk $S_n$ on $\mathbb{Z}$ at the first-passage time $N_d=\inf\{n\ge 0:S_n=d\}$, with $p=P[X_1=+1]\in[1/2,1)$. Using generating-function techniques and integral representations, we derive explicit formulas for the expectation and variance of $R_{N_d}/N_d$ along with monotonicity properties in the threshold $d$ and the bias $p$. We also provide closed-form expressions and use them to design unbiased coin-flipping estimators of $\pi$ based on first-passage sampling; the resulting schemes illustrate how biasing the coin can dramatically improve both approximation accuracy and computational cost.
From: Davy Paindaveine [view email]
[v1]
Wed, 24 Dec 2025 16:05:15 UTC (88 KB)
[v2]
Fri, 26 Dec 2025 10:07:03 UTC (86 KB)
[v3]
Mon, 31 Aug 2026 22:36:10 UTC (90 KB)
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